Between Sobolev and Poincaré

By Rafal Latala, K. Oleszkiewicz. 2000

171 citations recorded by the source corpus.

Let a a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let μ(dx)=c r n exp(-(|x1| r +|x2| r +...+|x n | r ))dx1dx2...dx n be a probability measure on the Euclidean space (R n , ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R n and any p ∈ [1,2) $$E_\mu f^2 - (E_\mu \left| f \right|^p )^{2/p} \leqslant C(2 - p)^a E_\mu \left\| {\nabla f} \right\|^2$$ . We prove also that if for some probabilistic measure μ on R n the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R n → R such that |h(x)-h(y)|≤∥x-y∥ there is E μ |h| < ∞ and $$\mu (h - E_\mu h > \sqrt C \cdot t) \leqslant e^{ - Kt^r }$$ for t > 1, where K > 0 is some universal constant.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/11c5fb4541180b7c293ca33d5be2139dc76aa5ab

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In Applied Mathematics

Citations in the corpus

Modified logarithmic Sobolev inequalities and transportation inequalities (2004, methodology): “In Section 3 we prove, as in [ LO00 ], that the measure � � defined in (2) satisfies the inequality LSIa,�(C).”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “Previous works of Barthe–Roberto [13], Barthe–Cattiaux–Roberto [9] and Roberto–Zegarlinski [84] have shown that 2-capacity inequalities are often equivalent to certain other types of functional inequalities, such as the Lata la–Oleszkiewicz inequality [60] (or more general Beckner-type inequalities)”
Displacement convexity of generalized relative entropies (2010, methodology): “Note that, if (M,ω) satisfies RicN ≥ K for some K ∈ R and N ∈ [n,∞), then it behaves like a Riemannian manifold with dimension bounded above by N and Ricci curvature bounded below by K (see [Qi], [Lo], as well as [St2], [St3], [LV1], [LV2], [Vi2, Part III] related to the curvature-dimension conditio”
Stability Results for Logarithmic Sobolev and Gagliardo–Nirenberg Inequalities (2014, methodology): “The proof relies on an idea that can be found in Lata la and Oleszkiewicz [2000] and goes as follows.”
Improving (2018, unknown): “Thus we argue that these rainfall forecasts provide skill complementary to that from the hydrological initial conditions.”
Sobolev inequalities for probability measures on the real line (2003, background): “However, these constants degenerate when p tends to 2, so these results are not precise enough to address the inequalities of Latała and Oleszkiewicz.”
On the role of convexity in functional and isoperimetric inequalities (2008, methodology): “These include Beckner-type inequalities [6] (including the Lata la–Oleszkiewicz inequality as in [ 31 ]) and additive Φ-Sobolev inequalities [46].”
Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels (2014, background): “In general, an induction argument can be used to show that subadditivity is equivalent to the following convexity property [39]: for any two probability spaces ( X 1 , ν 1 ) and ( X 2 , ν 2 ) and any function f : where X 1 ∼ ν 1 .”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, methodology): “It was considered by LataÃla and Oleszkiewicz [36] for T (r) = C r.”
Isoperimetry between exponential and Gaussian (2006, methodology): “A remarkable feature of Beckner type inequalities (3) is the tensorization property: if „1 and „2 both satisfy (3) with constant C, then so does „1 › „2 [ 29 ].”
Concentration for independent random variables with heavy tails (2005, background): “[10]) yields dimension free Gaussian concentration, whereas recent inequalities devised by Lata la and Oleszkiewicz [ 9 ] provide intermediate rates, see also [4, 15, 3]. Note that these results only concern distributions with exponential or faster decay.”
Moment inequalities for functions of independent random variables (2005, methodology): “As pointed out in [25], provided that φ” is strictly positive, the condition 1/φ” concave is necessary for the tensorization property to hold.”